Tensors

There are several approaches to define tensors.

If we are consider finite dimensional vector spaces, then there is a very natural definition for tensors as multilinear maps.
Definition
Let $(V, +, \cdot)$ be a vector space. A $(r, s)$ -tensor, $T$ , over $V$ is a multilinear map

T : \underset{r -terms}{\underset{⏟}{V^{*} \times . . . \times V^{*}}} \times \underset{s -terms}{\underset{⏟}{V \times . . . \times V}} \mapsto R .

$◼$
Others flip the definition of an $(r, s)$ -tensor, in the sense that $r$ tells you how many $V$ terms appear in the above map and $s$ tells you how many $V^{*}$ terms appear there. It is important to make sure you know which convention you are dealing with before moving forward.

Better see tensor product.

In the case of smooth manifolds we have tensor fields...

Invariants of a tensor

An invariant of a tensor is a quantity associated with the tensor that remains unchanged under certain transformations of the coordinate system. Specifically, in the context of tensors, an invariant is a scalar that does not depend on the particular choice of basis or coordinate system used to describe the tensor.

In more detail:

For a tensor to have an invariant, there must exist some scalar function of the components of the tensor that remains the same when you perform a coordinate transformation (e.g., a rotation, reflection, or Lorentz transformation).
Invariants are crucial because they represent physical quantities that do not depend on the observer's point of view or the coordinate system used, making them physically meaningful.

For example, in the case of rank-(1,1) tensors (such as the Cauchy stress tensor or the shape operator) have common invariants like the trace (sum of the diagonal elements), determinant, and the eigenvalues of the tensor.

Another example second fundamental form#Invariants of the second fundamental form.